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Cremona group

The group $\def\Cr#1{{\rm Cr}(\mathbb{P}_k^#1)} \Cr{n}$ of birational automorphisms of a projective space $\mathbb{P}_k^n$
over a field $k$; or, equivalently, the group of Cremona
transformations (cf.
Cremona transformation) of $\mathbb{P}_k^n$.

The group ${\rm PGL}(n+1,k)$ of projective transformations of $\mathbb{P}_k^n$ is contained in a
natural manner in $\Cr{n}$ as a subgroup; when $n\ge 2$ it is a proper
subgroup. The group $\Cr{n}$ is isomorphic to the group ${\rm Aut}\; k(x_1,\dots,x_n)$ of
automorphisms over $k$ of the field of rational functions in $n$
variables over $k$. The fundamental result concerning the Cremona
group for the projective plane is Noether's theorem: The group $\Cr{2}$
over an algebraically closed field is generated by the quadratic
transformations or, equivalently, by the standard quadratic
transformation and the projective transformations (see
[1],
[7]); relations between these generators can be found
in

(see also ). It is not known to date (1987) whether the Cremona group
is simple. There is a generalization of Noether's theorem to the case
in which the ground field $k$ is not algebraically closed (see
[6]).

One of the most difficult problems in birational geometry is that of
describing the structure of the group $\Cr{3}$, which is no longer
generated by the quadratic transformations. Almost all literature on
Cremona transformations of three-dimensional space is devoted to
concrete examples of such transformations. Finally, practically
nothing is known about the structure of the Cremona group for spaces
of dimension higher than 3.

An important direction of research in the theory of Cremona groups is
the investigation of subgroups of $\Cr{n}$. The finite subgroups of $\Cr2$,
with $k$ algebraically closed, have been described up to conjugacy
(see
[8], and also
[6]). A classification of all involutions in $\Cr{2}$ was
obtained as far back as 1877 by E. Bertini (see e.g.
[4],
[5]). The question of describing all involutions in
$\Cr{n}$, $n\ge3$, remains open. All maximal connected algebraic subgroups in
$\Cr{2}$ were described by F. Enriques in 1893 (see
[4]): They are just the automorphism groups of all
minimal models of rational surfaces, i.e. of the plane $\mathbb{P}_k^2$, the
quadric $\mathbb{P}^1\times\mathbb{P}^1$ and of the series of ruled surfaces $\mathbb{P}_N$, $n\ge 2$. There are a
few generalizations of this result (see
[3],
[9]) to the case of the group $\Cr{n}$, $n\ge 3$.